How accumulation and slope undo each other

Mathematics · Calculus & Analysis · ages 19-20

Name ______________________ Date ____________

Evaluate the definite integral from 0 to 2 of (3x^2 + 1) dx.
Answer: ______________
Let g(x) = the integral from 0 to x of sin(t) dt. What is g'(x)?
- sin(x)
- -cos(x)
- cos(x)
- 0
With g of x as the integral from 0 to x of sin of t dt, what is g prime of x?
- sin of x
- minus cos of x
- 0
Evaluate the integral from 2 to 5 of 2x dx.
- 10
- 21
- 49
With F of x as the integral from 1 to x of 3t squared plus 2 dt, what is F prime?
- 6x
- x cubed plus 2x
- 3x squared plus 2
True or false: when you evaluate a definite integral with an antiderivative, it does not matter which antiderivative you pick, because the added constant cancels out.
Circle one: True False
Evaluate the definite integral from 0 to 3 of (2x + 1) dx.
Answer: ______________
Evaluate the definite integral from 0 to 1 of (6x^2 + 4x) dx.
Answer: ______________
Find d/dx of the integral from 0 to x^2 of cos(t) dt.
- cos(x^2)
- 2x cos(x^2)
- -sin(x^2)
- cos(2x)
What is d dx of the integral from 0 to x squared of cos of t dt?
- 2x cos of x squared
- minus sin of x squared
- cos of x squared

Answer key

For grown-ups. Fold this page away before handing over the rest.

10 · Using the Fundamental Theorem of Calculus Part 2, find an antiderivative and subtract its values at the two limits.
sin(x) · The first part of the Fundamental Theorem of Calculus says that differentiating an accumulation function just gives back the integrand, with x in place of t.
sin of x · Part 1 returns the integrand with x in place of t.
21 · Antiderivative x squared gives 25 minus 4.
3x squared plus 2 · A variable top limit differentiates to the integrand at that limit.
True · Any two antiderivatives differ by a constant C, and (F(b) + C) minus (F(a) + C) leaves F(b) minus F(a). That is why the Fundamental Theorem of Calculus works with any antiderivative.
12 · Antidifferentiate, then plug in the top and bottom limits and subtract, as the Fundamental Theorem of Calculus Part 2 allows.
4 · An antiderivative turns a hard limit-of-sums problem into simple substitution, thanks to the Fundamental Theorem of Calculus Part 2.
2x cos(x^2) · When the upper limit is a function of x, you use the Fundamental Theorem of Calculus together with the chain rule: evaluate the integrand at the upper limit, then multiply by the derivative of that limit.
2x cos of x squared · Evaluate at the top, then multiply by the top derivative 2x.